And specifically hot water for his feet Bet Hillel permitted, but for his entire body, no, as we learnt {tnan -- See Rif here}:

{MISHNA:IT ONCE HAPPENED THAT THE PEOPLE OF TIBERIAS DID THUS: THEY CONDUCTED A PIPE OF COLD WATER THROUGH AN ARM OF THE HOT SPRINGS.}SAID THE SAGES TO THEM: {IF ON THE SABBATH {the water is drawn through the pipe}, IT IS LIKE HOT WATER HEATED ON THE SABBATH, AND IS FORBIDDEN BOTH FOR WASHING AND FOR DRINKING;}IF ON A FESTIVAL, IT IS LIKE WATER HEATED ON A FESTIVAL, WHICH IS FORBIDDEN FOR WASHING BUT PERMITTED FOR DRINKING.

{which seems against Bet Hillel, but no, they were talking there about washing the entire body}

And in terms of washing on Yom Tov with water that had been heated from erev Yom Tov, we see that the Gaon writes that this that we say that a man may not wash his entire body on Shabbat with hot water that had been heated from erev Shabbat, that is specifically by Shabbat, but on Yom Tov it is permitted for him to wash his entire body with hot water that had been heated from erev Yom Tov, for it is not something that one is liable for shevut {mandated rest} nor reshut {an optional laudatory thing} nor a mitzvah, but rather it is a decree {because of what might happen} and when they decreed it was for Shabbat but not for Yom Tov.

In three things, Rabban Gamliel was strict like the words of Bet Shammai:We may not insulate hot water from Yom Tov to Shabbat.And we may not straighten the candlestick on Yom Tov.And we may not bake thick loaves of bread but rather thin.

Rabban Gamliel said: In all the days of my father's household they did not bake thick loaves of bread but rather thin.They said to him: What shall we do with your father's household? They were strict upon themselves and lenient on all of Israel to be able to back both thick and thin loaves.{Note: The Mishna in our gemara has here chararin, coal baked bread, rather than rekikin, wafers.}